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Holomorphic functional calculus

The holomorphic functional calculus, also known as the Riesz-Dunford functional calculus, provides a method in functional analysis to define the image of a bounded linear operator T on a complex Banach space under a holomorphic function f whose domain contains the spectrum \sigma(T) of T.^[1] This is achieved via the contour integral formula f(T) = \frac{1}{2\pi i} \int_\Gamma f(z) (zI - T)^{-1} \, dz, where \Gamma is a rectifiable Jordan curve in the domain of f that encloses \sigma(T) but lies in the resolvent set \rho(T).^[2] The resolvent operator (zI - T)^{-1} is analytic in z \in \rho(T), ensuring the integral is well-defined and independent of the choice of contour as long as it satisfies these conditions.^[1] This calculus extends the classical polynomial functional calculus, where polynomials in T are defined by power series, to the broader class of holomorphic functions, thereby facilitating the application of complex analysis tools to operator theory.^[2] A key property is the spectral mapping theorem, which states that \sigma(f(T)) = f(\sigma(T)), linking the spectrum of the resulting operator directly to the image of the original spectrum under f.^[1] Additionally, the map f \mapsto f(T) is an algebra homomorphism from the algebra of holomorphic functions on a neighborhood of \sigma(T) to the Banach algebra of operators, preserving addition, multiplication, and scalar multiplication.^[2] Originally formulated for bounded operators, the holomorphic functional calculus has been generalized to unbounded closed operators with nonempty resolvent sets, using similar contour integrals adjusted for the extended spectrum including infinity.^[2] Extensions include the H^\infty functional calculus for sectorial operators, which bounds the norm of f(T) in terms of the H^\infty norm of f on suitable sectors, aiding in the study of evolution equations and semigroups.^[3] In the context of C^*-algebras and normal operators on Hilbert spaces, it aligns with the continuous functional calculus via spectral measures, enabling applications in quantum mechanics, partial differential equations, and approximation theory.^[1]

Motivation

Need for a general functional calculus

The polynomial functional calculus serves as the foundational approach for applying functions to bounded linear operators on Banach spaces, where a polynomial p(z) = \sum_{k=0}^n a_k z^k is mapped to the operator p(A) = \sum_{k=0}^n a_k A^k, with powers defined recursively via composition and the identity operator for A^0.^[4] This method extends naturally to power series for entire functions, such as the exponential, by term-by-term application when the series converges in the operator norm.^[4] However, polynomials are inherently limited to finite-degree expressions, preventing direct representation of non-polynomial analytic functions like the exponential e^z or logarithm \log z without approximation, which lacks uniformity over the operator's spectrum and fails to preserve key algebraic properties for general holomorphic functions.^[4] These limitations motivated the development of a more general framework in the mid-20th century, particularly to address challenges in spectral theory where operators require evaluation under arbitrary holomorphic functions defined on neighborhoods of their spectra. Nelson Dunford and Jacob T. Schwartz introduced the holomorphic functional calculus in their seminal work, providing a systematic extension that incorporates complex analysis to define f(A) for holomorphic f, ensuring consistency with the polynomial case and enabling broader applications in operator theory. A primary application arises in solving linear differential equations with operator coefficients, such as the abstract evolution equation \frac{d}{dt} x(t) = Ax(t), where the solution involves the evolution operator e^{tA}, which the holomorphic calculus defines rigorously even when A is non-normal and polynomials alone cannot suffice.^[5] Similarly, for time-dependent problems like the Schrödinger equation, the unitary evolution operator e^{itA} (with t real) leverages the calculus to propagate initial states while preserving spectral properties essential for quantum mechanical interpretations.^[6]

Role of the spectrum in functional calculus

In functional analysis, the spectrum of a bounded linear operator T on a Banach space X, denoted \sigma(T), is defined as the set of all complex numbers \lambda \in \mathbb{C} such that T - \lambda I is not invertible in the algebra B(X) of bounded operators on X.^[7] This set captures the values of \lambda for which the operator equation (T - \lambda I)x = y fails to have a unique solution for every y \in X, generalizing the notion of eigenvalues from finite-dimensional spaces.^[8] The resolvent set \rho(T) is the complement \mathbb{C} \setminus \sigma(T), which is an open subset of the complex plane.^[9] For \lambda \in \rho(T), the resolvent operator R(\lambda, T) = (\lambda I - T)^{-1} exists as a bounded linear operator on X and is holomorphic as a function of \lambda in \rho(T).^[10] This resolvent function serves as a fundamental tool in spectral theory, providing analytic continuation properties that underpin extensions to operator functions.^[9] For bounded operators, the spectrum \sigma(T) is always a non-empty compact subset of \mathbb{C}, meaning it is closed and bounded.^[7] The boundedness follows from the spectral radius formula, which states that the radius r(T) = \sup \{ |\lambda| : \lambda \in \sigma(T) \} \leq \|T\|, ensuring \sigma(T) lies within the disk of radius \|T\| centered at the origin.^[9] The closedness arises from the continuity of the resolvent in \rho(T), preventing accumulation points outside \sigma(T).^[11] In the context of holomorphic functional calculus, the spectrum \sigma(T) plays a crucial role by constraining the domain of applicable holomorphic functions f. Specifically, f must be holomorphic on an open neighborhood of \sigma(T) to ensure that the resulting operator f(T) is well-defined, as this holomorphy guarantees that f remains analytic and bounded across the spectral region, avoiding any singularities that could arise within \sigma(T).^[9] The compactness of \sigma(T) facilitates this by allowing contours to enclose the entire spectrum compactly, providing a prerequisite for the calculus's consistency and analytic properties.^[7] This spectral constraint ensures that the functional calculus extends naturally from polynomials to a broader class of holomorphic functions while preserving operator-theoretic structure.^[9]

Definition for bounded operators

Resolvent function and Cauchy integral formula

The resolvent function of a bounded linear operator T on a complex Banach space X is given by R(\lambda, T) = (\lambda I - T)^{-1}, where \lambda \in \mathbb{C} belongs to the resolvent set \rho(T), the complement of the spectrum \sigma(T). This operator-valued function plays a central role in the holomorphic functional calculus, as it allows the extension of scalar holomorphic functions to operators via contour integration.^[12] Given a function f holomorphic in an open neighborhood of \sigma(T), the holomorphic functional calculus defines f(T) using the Cauchy integral formula adapted to operators:

f(T) = \frac{1}{2\pi i} \int_\Gamma f(\lambda) R(\lambda, T) \, d\lambda,

where \Gamma is a simple closed positively oriented contour lying entirely in \rho(T) and enclosing \sigma(T) in its interior, with f holomorphic on and inside \Gamma. This construction generalizes the scalar Cauchy integral formula f(z) = \frac{1}{2\pi i} \int_\Gamma \frac{f(\lambda)}{\lambda - z} \, d\lambda by replacing the scalar kernel (\lambda - z)^{-1} with the resolvent R(\lambda, T).^[13]^[12] The integral on the right-hand side is well-defined as a Bochner integral in the Banach space of bounded linear operators on X, or equivalently as the norm limit of Riemann sums approximating the contour integral. This requires that the operator-valued function \lambda \mapsto f(\lambda) R(\lambda, T) be continuous (hence bounded) on \Gamma, which follows from the holomorphy of f in a neighborhood of \Gamma and the fact that R(\cdot, T) is holomorphic on \rho(T). The choice of \Gamma is flexible as long as it satisfies these conditions, ensuring the integral is independent of the specific contour.^[13]

Analyticity and Neumann series expansion

The holomorphic functional calculus defines the operator f(T) for a bounded linear operator T on a Banach space and a function f holomorphic in a neighborhood of the spectrum \sigma(T) via the contour integral formula

f(T) = \frac{1}{2\pi i} \int_\Gamma f(z) R(z, T) \, dz,

where R(z, T) = (zI - T)^{-1} is the resolvent and \Gamma is a positively oriented contour enclosing \sigma(T).^[14]^[9] The map sending T to f(T) is analytic in the operator norm topology when f is fixed and holomorphic in a suitable region. This analyticity follows from differentiating under the integral sign, justified by the dominated convergence theorem for Bochner integrals in Banach spaces, combined with resolvent estimates such as \|R(z, T)\| \leq \dist(z, \sigma(T))^{-1}.^[14]^[9] Specifically, for a differentiable path T(t) with t in a neighborhood of 0, the derivative \frac{d}{dt} f(T(t)) = f'(T(t)) T'(t) holds, with the integral representation ensuring uniform boundedness on compact sets away from the spectrum.^[9] A key tool for establishing this analyticity is the Neumann series expansion of the resolvent. For |\lambda| > \|T\|, the resolvent admits the series representation

R(\lambda, T) = \sum_{n=0}^\infty \lambda^{-n-1} T^n = -\frac{1}{\lambda} \sum_{n=0}^\infty \left( \frac{T}{\lambda} \right)^n,

which converges in the operator norm since \|T/\lambda\| < 1.^[14]^[9] This expansion extends locally around any \mu \in \rho(T) via R(\lambda, T) = \sum_{n=0}^\infty (\mu - \lambda)^n R(\mu, T)^{n+1} for |\lambda - \mu| < \|R(\mu, T)\|^{-1}, providing an analytic continuation of the resolvent map \lambda \mapsto R(\lambda, T) to the resolvent set \rho(T).^[14] These series facilitate resolvent estimates essential for interchanging differentiation and integration in the functional calculus.^[9] When f admits a power series expansion f(z) = \sum_{n=0}^\infty a_n z^n convergent in a disk D_r(0) with radius r > 0, the functional calculus extends this to operators via f(T) = \sum_{n=0}^\infty a_n T^n, provided \sigma(T) \subseteq D_r(0).^[14]^[9] The series converges in the operator norm, with the radius of convergence tied to the spectral radius r(T) = \sup \{ |\lambda| : \lambda \in \sigma(T) \}, ensuring absolute convergence when r > r(T).^[14] This representation aligns with the contour integral by deforming the contour to a circle around 0, leveraging the Neumann series for the resolvent outside \sigma(T).^[9] For bounded operators, the spectrum \sigma(T) is always compact, which implies uniform bounds on the resolvent along suitable contours enclosing \sigma(T). Specifically, on a contour \Gamma at a positive distance d > 0 from the compact set \sigma(T), the estimate \|R(z, T)\| \leq d^{-1} holds uniformly for z \in \Gamma.^[14]^[9] This uniformity ensures the contour integral defining f(T) is well-behaved, with \|f(T)\| \leq \frac{\length(\Gamma)}{2\pi} \cdot \max_{z \in \Gamma} |f(z)| \cdot d^{-1}, facilitating analytic dependence and stability under perturbations of T.^[14]

Well-definedness of the functional calculus

To establish the well-definedness of the holomorphic functional calculus for a bounded linear operator T on a complex Banach space, consider a function f that is holomorphic on an open set U containing the spectrum \sigma(T). The functional calculus defines f(T) via the contour integral \frac{1}{2\pi i} \int_{\Gamma} f(\lambda) R(\lambda, T) \, d\lambda, where R(\lambda, T) = (\lambda I - T)^{-1} is the resolvent operator and \Gamma is a closed contour in U \setminus \sigma(T) that encloses \sigma(T) in its positive orientation.^[9] A preliminary fact ensures the independence of this definition on the specific contour choice. For any two such homologous contours \Gamma_1 and \Gamma_2, both enclosing \sigma(T), the difference \int_{\Gamma_1} R(\lambda, T) \, d\lambda - \int_{\Gamma_2} R(\lambda, T) \, d\lambda = 0. This follows from Cauchy's theorem applied to the resolvent, which is holomorphic in the open resolvent set \mathbb{C} \setminus \sigma(T), with no singularities inside the region bounded by \Gamma_1 - \Gamma_2.^[9] The main argument for well-definedness proceeds by contour deformation. Given two valid contours \Gamma_1 and \Gamma_2, deform \Gamma_1 continuously to \Gamma_2 within U \setminus \sigma(T), avoiding the compact set \sigma(T). During this deformation, the integrand f(\lambda) R(\lambda, T) remains holomorphic in the deformed region, as f is holomorphic on U and the resolvent is analytic outside \sigma(T). By the global Cauchy theorem for vector-valued holomorphic functions, the integral remains invariant under such deformations, yielding \int_{\Gamma_1} f(\lambda) R(\lambda, T) \, d\lambda = \int_{\Gamma_2} f(\lambda) R(\lambda, T) \, d\lambda. The assumption that f is holomorphic on an open set containing \sigma(T) guarantees no poles or singularities of the integrand inside the contours.^[9] This invariance resolves potential ambiguities in the definition, confirming that f(T) is uniquely determined regardless of the choice of contour \Gamma, provided it satisfies the enclosing and holomorphy conditions.^[9]

Fundamental properties

Polynomial approximation and homomorphism

The holomorphic functional calculus recovers the classical polynomial functional calculus as a special case. For a polynomial p(z) = \sum_{k=0}^n a_k z^k and a bounded linear operator T on a complex Banach space, the polynomial calculus defines p(T) = \sum_{k=0}^n a_k T^k, where T^k denotes the k-fold composition of T with itself (and T^0 = I, the identity operator).^[9] This construction forms a unital algebra homomorphism from the ring of polynomials to the algebra of bounded operators.^[15] The Riesz-Dunford calculus, defined via the Cauchy integral formula over a contour enclosing the spectrum \sigma(T), agrees with this polynomial definition when applied to polynomials, thereby extending it to general holomorphic functions.^[9] A key feature of the Riesz-Dunford calculus is its homomorphism property under function composition. Let \Phi denote the calculus map, which assigns to each holomorphic function f defined on an open set containing \sigma(T) the operator \Phi(f) = f(T). If g is holomorphic on a neighborhood of \sigma(T) such that g(\sigma(T)) \subset \Omega_f, where \Omega_f is a domain containing \sigma(g(T)) on which f is holomorphic, then \Phi(f \circ g) = \Phi(f) \circ \Phi(g), or equivalently, f(g(T)) = (f \circ g)(T).^[16] This composition rule holds under these domain conditions and reflects the spectral compatibility of the functions involved.^[16] The Riesz-Dunford calculus serves as the natural holomorphic extension of the polynomial calculus because polynomials are dense in the space of holomorphic functions on a fixed domain, with respect to the topology of uniform convergence on compact subsets.^[9] This density, combined with the continuity of the calculus map, ensures that the extension is well-defined and preserves the algebraic structure of the polynomial case.^[9] For instance, consider the function f(z) = z^{-1}, which is holomorphic on \mathbb{C} \setminus \{0\}. If $0 \notin \sigma(T), then f(T) = T^{-1}, the inverse of T, as the Cauchy integral representation yields the negative of the resolvent at 0, which is T^{-1}. This example illustrates how the calculus handles poles outside the spectrum, generalizing the polynomial approach to resolvents and inverses.

Continuity under compact convergence

The holomorphic functional calculus exhibits continuity with respect to compact convergence of the defining functions. Let T be a bounded linear operator on a complex Banach space with spectrum \sigma(T) \subset G, where G \subset \mathbb{C} is open. If a sequence of functions \{f_n\} holomorphic on G converges to a holomorphic function f on G uniformly on every compact subset of G (i.e., compact convergence), then f_n(T) \to f(T) in the operator norm: \|f_n(T) - f(T)\| \to 0 as n \to \infty. This continuity follows from the Cauchy integral representation of the functional calculus. For a rectifiable contour \Gamma in G that encloses \sigma(T) in its interior and lies in the resolvent set \rho(T), the operator f(T) is given by

f(T) = \frac{1}{2\pi i} \int_\Gamma f(z) (zI - T)^{-1} \, dz,

where the integral is interpreted in the Bochner sense for Banach space-valued functions. Since \sigma(T) is compact, one may choose \Gamma such that the distance from \Gamma to \sigma(T) is positive, ensuring \|(zI - T)^{-1}\| \leq M for some constant M > 0 and all z \in \Gamma. Under compact convergence, f_n \to f uniformly on the compact set \Gamma, so the integrands f_n(z) (zI - T)^{-1} converge uniformly to f(z) (zI - T)^{-1} on \Gamma. The length of \Gamma is finite, yielding

\left\| \int_\Gamma [f_n(z) - f(z)] (zI - T)^{-1} \, dz \right\| \leq \frac{\text{length}(\Gamma)}{2\pi} \cdot \sup_{z \in \Gamma} |f_n(z) - f(z)| \cdot M \to 0

as n \to \infty, by the triangle inequality for integrals. To handle the general case, contours may be chosen arbitrarily close to \sigma(T) while maintaining uniform bounds on the resolvent via the maximum modulus principle applied to the holomorphic resolvent function on components of G \setminus \sigma(T).^[17] This topological continuity has key implications for approximation within the functional calculus. Holomorphic functions on G can be uniformly approximated on compact neighborhoods of \sigma(T) by rational functions with poles outside G (via Runge's theorem), and by polynomials under suitable conditions on the domain (e.g., when the complement of the compact neighborhood is connected), and the continuity ensures that f(T) is the norm limit of the corresponding rational or polynomial evaluations [at

T](/page/AT&T)

.^[9] Thus, the functional calculus extends the polynomial calculus holomorphically while preserving limits under compact convergence. Unlike pointwise convergence of \{f_n\} to f, which may fail to imply \|f_n(T) - f(T)\| \to 0 due to lack of uniform control over the resolvent integrals on contours, compact convergence is essential to bound the operator norms effectively. Pointwise limits might only yield strong or weak convergence of f_n(T) to f(T), but not necessarily in norm.

Uniqueness theorem

The uniqueness theorem establishes that the holomorphic functional calculus defined via the Cauchy integral formula is the unique extension of the polynomial functional calculus to the algebra of holomorphic functions on an open set containing the spectrum of a bounded linear operator T on a Banach space X. Specifically, let \sigma(T) denote the spectrum of T, and let U \subset \mathbb{C} be open with \sigma(T) \subset U. Any map \Phi: \mathrm{Hol}(U) \to B(X) that extends the polynomial functional calculus (i.e., \Phi(p) = p(T) for every polynomial p), is \mathbb{C}-linear, unital (with \Phi(\mathrm{id}) = T), multiplicative, and continuous with respect to the compact-open topology on \mathrm{Hol}(U) must coincide with the Cauchy integral map \Phi(f) = \frac{1}{2\pi i} \int_\Gamma f(z) (zI - T)^{-1} \, dz, where \Gamma is a positively oriented contour in U enclosing \sigma(T). This result underscores the canonical nature of the construction, ensuring that the functional calculus is intrinsically tied to the spectral properties of T.^[4] The proof proceeds by leveraging the density of certain approximating functions and the continuity of \Phi. First, note that the resolvent R(\lambda, T) = (\lambda I - T)^{-1} uniquely determines \Phi on rational functions with poles outside U, since such functions can be expressed in terms of resolvents, and the polynomial case follows immediately. By Runge's theorem, every f \in \mathrm{Hol}(U) can be uniformly approximated on compact subsets of U by sequences of such rational functions (or equivalently, by polynomials when U is simply connected). Given the continuity of \Phi under compact convergence, \Phi(f) is then the limit of \Phi applied to these approximants, which uniquely matches the Cauchy integral representation. This approximation argument directly implies that any two such maps \Phi and \Psi satisfy \Phi(f) = \Psi(f) for all f \in \mathrm{Hol}(U).^[4] A Liouville-type argument provides additional insight in the special case where U = \mathbb{C} and the functional calculus extends to entire functions, as occurs for bounded operators via power series expansion. Suppose \Phi and \Psi are two such extensions that agree on polynomials. Their difference D(f) = \Phi(f) - \Psi(f) vanishes on all polynomials, which are dense in the space of entire functions under the compact-open topology. By continuity, D(f) = 0 for all entire f. Moreover, if one considers the generating entire function associated with the difference (e.g., via the formal power series), the identity theorem for holomorphic functions—analogous to Liouville's theorem in its uniqueness of analytic continuation—ensures that any entire function vanishing on the non-empty open set of polynomial evaluations must be identically zero, reinforcing the uniqueness. This perspective highlights how the theorem precludes non-trivial alternative extensions even in the global case.^[9] As a consequence, the holomorphic functional calculus is canonical, independent of the choice of contour \Gamma (as long as it satisfies the required conditions), and serves as the foundational tool for spectral theory without ambiguity in its definition or properties. This uniqueness guarantees that applications, such as spectral mappings or projections, inherit a well-defined structure directly from the operator's resolvent.

Spectral applications

Spectral mapping theorem

The spectral mapping theorem is a fundamental result in the holomorphic functional calculus for bounded linear operators on complex Banach spaces. It establishes a precise relationship between the spectrum of an operator and the spectrum of its image under a holomorphic function. Specifically, let T be a bounded linear operator on a complex Banach space X, and let f be a function holomorphic on an open neighborhood U \subseteq \mathbb{C} containing the spectrum \sigma(T). Then the holomorphic functional calculus defines f(T), and the theorem states that

\sigma(f(T)) = f(\sigma(T)).

This equality holds because f has no singularities in U, ensuring the mapping is well-behaved without additional exceptional sets arising from poles.^[9] To prove the theorem, consider the two inclusions separately, relying on the contour integral representation of the functional calculus: for a positively oriented contour \Gamma in U enclosing \sigma(T) but no other points of the spectrum,

f(T) = \frac{1}{2\pi i} \int_\Gamma f(\mu) (\mu I - T)^{-1} \, d\mu.

First, show f(\sigma(T)) \subseteq \sigma(f(T)). Suppose \lambda \in f(\sigma(T)), so \lambda = f(\mu_0) for some \mu_0 \in \sigma(T). Assume for contradiction that \lambda \in \rho(f(T)), so f(T) - \lambda I is invertible. However, since the functional calculus is a homomorphism (as established in the polynomial approximation and composition properties), and polynomials satisfy the spectral mapping property, the density of polynomials in the holomorphic functions under uniform convergence on compact sets implies that f(T) - f(\mu_0) I cannot be invertible, leading to a contradiction. More directly, the non-invertibility at \mu_0 propagates through the mapping.^[9] For the reverse inclusion \sigma(f(T)) \subseteq f(\sigma(T)), let \lambda \notin f(\sigma(T)). Then f(\mu) - \lambda \neq 0 for all \mu \in \sigma(T), and since \sigma(T) and f(\sigma(T)) are compact, \mathrm{dist}(\lambda, f(\sigma(T))) > 0. The function k(\mu) = \frac{1}{f(\mu) - \lambda} is thus continuous and bounded on \sigma(T). The holomorphic functional calculus extends the continuous functional calculus for functions continuous on \sigma(T), so k(T) (f(T) - \lambda I) = I and (f(T) - \lambda I) k(T) = I, showing that f(T) - \lambda I is invertible with bounded inverse k(T). This uses the analyticity of f to ensure consistency with the integral representation.^[9] If f had poles within U, the equality might fail, with potential additional points in \sigma(f(T)) from residues at poles, but the holomorphy assumption on a neighborhood of \sigma(T) precludes this, yielding the exact mapping. An illustrative example is the exponential function f(z) = e^z, which is entire, so for any bounded T, \sigma(e^T) = e^{\sigma(T)}; this follows directly from the theorem and is useful in applications like semigroup theory.^[9]

Spectral projections and invariant subspaces

In the holomorphic functional calculus for a bounded linear operator T on a complex Banach space X, spectral projections are constructed for Borel subsets E \subseteq \sigma(T) using contour integrals over the resolvent. Specifically, choose an open set \Omega containing E \cap \sigma(T) such that \partial \Omega lies in the resolvent set \rho(T), and define

P_E = \frac{1}{2\pi i} \int_{\partial \Omega} (z I - T)^{-1} \, dz,

where the contour is positively oriented.^[9] This formula arises from applying the calculus to the characteristic function \chi_E of E, extended holomorphically off the spectrum.^[9] The operator P_E is a bounded projection with range \operatorname{ran}(P_E) and kernel \ker(P_E) both closed subspaces of X. It satisfies idempotence P_E^2 = P_E, since \chi_E^2 = \chi_E under the functional calculus, and commutes with T, i.e., P_E T = T P_E, because the resolvent (z I - T)^{-1} commutes with T for each z \in \rho(T).^[9] Moreover, if \{E_j\}_{j \in J} is a finite or countable disjoint family of Borel sets whose union is a Borel set F \subseteq \sigma(T), then \sum_j P_{E_j} = P_F in the strong operator topology, and if the E_j partition \sigma(T), the sum equals the identity operator I.^[9] The spectrum of the restricted operator satisfies \sigma(T|_{\operatorname{ran}(P_E)}) = E \cap \sigma(T) and \sigma(T|_{\ker(P_E)}) = \sigma(T) \setminus (E \cap \sigma(T)). These projections induce invariant subspace decompositions of X. For a partition of \sigma(T) into connected components \{\Delta_j\}, the space decomposes as X = \bigoplus_j \operatorname{ran}(P_{\Delta_j}), where each \operatorname{ran}(P_{\Delta_j}) is invariant under T and the restriction T|_{\operatorname{ran}(P_{\Delta_j})} has spectrum \Delta_j.^[9] The Riesz decomposition theorem generalizes this to functions constant on spectral sets: for disjoint clopen subsets K_1, K_2 \subseteq \sigma(T) (in the hull-kernel topology), there exists a projection P such that X = \operatorname{ran}(P) \oplus \ker(P), with \sigma(T|_{\operatorname{ran}(P)}) \subseteq K_1 and \sigma(T|_{\ker(P)}) \subseteq K_2.^[9] This provides a direct sum decomposition into T-invariant subspaces corresponding to separated parts of the spectrum.

Functional calculus for unbounded operators

The holomorphic functional calculus extends naturally to closed densely defined unbounded operators T on a complex Banach space X, provided that the spectrum \sigma(T) is contained in a suitable open set where a holomorphic function f is defined, and a contour \Gamma can be chosen in the resolvent set \rho(T) to enclose \sigma(T). For such operators, the resolvent R(\lambda, T) = (\lambda I - T)^{-1} exists for \lambda \in \rho(T), and the functional calculus is defined via the Cauchy integral formula applied to vectors in X. Specifically, for x \in X, the action of f(T) on x is given by

f(T) x = \frac{1}{2\pi i} \int_\Gamma f(\lambda) R(\lambda, T) x \, d\lambda,

where the integral is interpreted in the strong sense, and \Gamma is a rectifiable oriented contour enclosing \sigma(T) counterclockwise. This definition parallels the bounded case but requires careful selection of \Gamma to ensure the integral converges, often assuming T is sectorial—meaning \sigma(T) \subset \overline{S}_\theta for some sector S_\theta = \{ z \in \mathbb{C} \setminus \{0\} : |\arg z| < \theta \} with \theta < \pi/2, and the resolvent satisfies suitable bounds \|R(\lambda, T)\| \leq M / |\lambda| for \lambda \in \mathbb{C} \setminus \overline{S}_\theta.^[21] The domain of the unbounded operator f(T), denoted \dom(f(T)), is the set of all x \in X for which the integral defining f(T)x converges absolutely, precisely

\dom(f(T)) = \left\{ x \in X : \int_\Gamma |f(\lambda)| \|R(\lambda, T) x\| \, |d\lambda| < \infty \right\}.

This domain is dense in X under the assumptions on T and f, and f(T) is a closed operator because the graph of f(T) is closed, as ensured by the closed graph theorem applied to the integral representation. For sectorial operators, the calculus is often developed within the space of bounded holomorphic functions H^\infty(S_\theta) on the sector, ensuring that f(T) remains sectorial with the same angle if f preserves the sector.^[22] The functional calculus retains key algebraic properties from the bounded setting where they are well-defined. In particular, it forms a homomorphism: if f and g are holomorphic on a common neighborhood of \sigma(T), then (f + g)(T) = f(T) + g(T) with \dom((f + g)(T)) = \dom(f(T)) \cap \dom(g(T)), and similarly for multiplication fg(T) = f(T) g(T) on the intersection of domains, provided the compositions make sense. The spectral mapping theorem holds in the form \sigma(f(T)) = f(\sigma(T)), where the spectrum of the possibly unbounded f(T) is taken relative to its domain, ensuring that the image under f captures the essential spectral behavior. This property is crucial for applications, such as defining functions of generators of analytic semigroups. Challenges arise primarily from the unbounded nature of \sigma(T), which prevents enclosing it with a compact contour like in the bounded case; instead, contours such as deformed sector boundaries or vertical lines in right half-planes are used, relying on resolvent estimates to control growth at infinity. For non-sectorial operators, the calculus may require additional restrictions, such as T having spectrum in a parabola or half-plane, to guarantee well-definedness and boundedness of f(T). These extensions highlight the need for operator classes like sectorial or strip-type operators to maintain the calculus's utility in spectral theory and evolution equations.^[9]

Dunford-Schwartz calculus and generalizations

The Dunford-Schwartz calculus extends the holomorphic functional calculus to a broader class of functions, particularly measurable functions, by employing spectral projections for normal operators on Hilbert spaces. For a bounded normal operator T on a Hilbert space H with spectral measure E, the functional calculus defines f(T) = \int_{\sigma(T)} f(\lambda) \, dE(\lambda) for bounded Borel measurable functions f: \sigma(T) \to \mathbb{C}, where the integral is understood in the strong operator topology.^[9] This construction generalizes the Riesz-Dunford integral representation and ensures compatibility with the holomorphic case, as the spectral integral coincides with the contour integral formula for holomorphic f.^[9] The calculus preserves key properties such as the spectral mapping theorem and provides a homomorphism from the algebra of bounded measurable functions on \sigma(T) to the von Neumann algebra generated by T.^[23] In the context of C*-algebras, the Dunford-Schwartz framework connects to the Gelfand transform for commutative cases, where a unital commutative C*-algebra A is isometrically isomorphic to C(\Delta(A)) via the Gelfand transform \hat{\cdot}: A \to C(\Delta(A)), with \Delta(A) the space of maximal ideals.^[24] For normal elements in commutative C*-algebras, this identifies the holomorphic functional calculus with pointwise application of holomorphic functions on the spectrum, yielding a *-homomorphism that extends the scalar case and aligns with the spectral theorem for normal operators.^[25] This linkage facilitates applications in spectral theory within commutative operator algebras, such as resolving functions of self-adjoint elements via continuous extensions.^[26] Further generalizations include the Sz.-Nagy-Foias functional calculus for contractions on Hilbert spaces, which employs the Hardy space H^\infty(\mathbb{D}) and unitary dilations to define f(T) for analytic functions f in the unit disk, ensuring a contractive homomorphism that extends beyond normal operators to those with defect operators. Another variant is the holomorphic functional calculus in several variables for commuting tuples of bounded operators, where functions holomorphic in multiple complex variables are applied via joint spectra and multivariable contour integrals, preserving algebra homomorphisms under joint holomorphy.^[27] Post-2000 developments have extended these ideas to non-commutative settings, such as noncommutative holomorphic functional calculi for tuples in non-commutative Banach algebras, using joint spectra and Frechet algebra presheaves to handle multivariable non-commutativity while maintaining analytic properties like the spectral mapping theorem.^[28] These extensions address limitations in classical frameworks for non-normal or multivariable operators, with applications in non-commutative geometry and quantum operator theory.^[29]

References

[1]
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Below is a merged summary of the holomorphic functional calculus based on the provided segments. To retain all the detailed information in a dense and organized manner, I will use a table in CSV format to consolidate the key aspects (Definition, Construction, Properties, Key Facts, and URLs) across the segments. Following the table, I will provide a concise narrative summary that integrates the overlapping and unique details.
[2]
[PDF] Notes on Banach Algebras and Functional Calculus - OSU Math
Apr 23, 2014 · Analytic functional calculus allows us to define analytic functions of oper-.
[3]
[PDF] Chapman University Digital Commons The H∞ Functional Calculus ...
Nov 24, 2015 · In the Splitting Lemma the two holomorphic functions F and G depend on the complex plane Ci that we consider. The Cauchy formula for slice ...
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[PDF] Holomorphic Functions of Bounded Operators
Now this is indeed a functional calculus, called the polynomial functional ... The functional calculus for entire functions works for every bounded operator.
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[PDF] Linear differential equations and functions of operators
Applications to PDEs typically involves functional calculus of unbounded differential operators D on a space like H = L2(Rn) (rather than bounded operators T).
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[PDF] Functional calculus for groups and applications to evolution equations
We show that Fattorini's theorem is related to the fact that arctan(A) is a bounded operator if −iA generates a. C0-group of type < 1 on a UMD space X. Section ...Missing: solving | Show results with:solving
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[PDF] Chapter 9: The Spectrum of Bounded Linear Operators
It is therefore necessary to define the spectrum of a linear operator on an infinite-dimensional space in a more general way than as the set of eigenvalues. We ...
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[PDF] Spectrum (functional analysis)
Mar 12, 2013 · In functional analysis, the concept of the spectrum of a bounded operator is a generalisation of the concept of eigenvalues for matrices.
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[PDF] Lectures on Functional Calculus
Mar 19, 2018 · The topic was “Functional Calculus” and it was my responsability to pro- vide the necessary text.
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[PDF] LECTURE 28 - Spectrum of bounded linear operators.
Definition and classification of spectrum. Def Let The L(X,X) where X is a Banach space,. λEC is called a regular point if π-xI is invertible, ie. Otherwise.
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[PDF] Spectrum of bounded operators
Nov 16, 2022 · 1.2 Spectrum of bounded operators - Resolvent. Let E be a Banach space. 1.2.1 Definition and basic properties. Definition 1.10. Let A P LpEq ...
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A Course in Functional Analysis - John B. Conway - Google Books
A Course in Functional Analysis Volume 96 of Graduate Texts in Mathematics. Author, John B. Conway. Edition, 2, illustrated, reprint. Publisher, Springer ...
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Linear Operators, Part 1: General Theory - Google Books
This classic text, written by two notable mathematicians, constitutes a comprehensive survey of the general theory of linear operations.<|control11|><|separator|>
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[PDF] LINEAR OPERATORS AND THEIR SPECTRA
This wide ranging but self-contained account of the spectral theory of non-self-adjoint linear operators is ideal for postgraduate students and researchers, ...
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[PDF] Holomorphic functional calculus for sectorial operators - TEMat
This holomorphic functional calculus is also called the Dunford–Riesz functional calculus. Note that the two previous examples are special cases of the ...Missing: history | Show results with:history
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[PDF] a general framework for holomorphic functional calculi
Abstract. We present an abstract approach to the construction of holomor- phic functional calculi for unbounded operators and apply it to the special case.
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The holomorphic functional calculus II (definition and basic properties)
Jan 5, 2014 · In this post we continue our discussion of the holomorphic functional calculus for elements of a Banach algebra (or operators).
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The Functional Calculus for Sectorial Operators - SpringerLink
The Functional Calculus for Sectorial Operators. Book; © 2006. 1st edition ... Markus Haase. Pages 1-17. The Functional Calculus for Sectorial Operators.
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[PDF] The Functional Calculus for Sectorial Operators and ... - Uni Ulm
... Holomorphic Semigroups 66 —. The Logarithm and the Imaginary Powers 69 ... Dunford-Riesz calculus, see [Con90,. Chapter VII,3 4]. This can be generalized ...
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Appendix: The Riesz–Dunford functional calculus
In this Appendix we collect some basic material on the Riesz–Dunford functional calculus useful for the readers who are not familiar with this subject. This ...
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[PDF] Lectures on C∗-algebras - arXiv
Oct 11, 2021 · Chapter 3 is mainly about the Gelfand transform and its consequences. So, the Gelfand transform on commutative Banach algebras and C∗-algebras ...
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[PDF] An introduction to Banach algebras and operator algebras
Apr 30, 2021 · This document introduces Banach algebras and operator algebras, focusing on their basic properties, and assumes prior knowledge of functional ...
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[PDF] Spectral Theory and the Gelfand Transform - Cornerstone
This thesis specifically explores this connection from unital Banach and C∗ algebras and their analytic representation via the Gelfand transform. Informally, ...
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Divided Differences and Multivariate Holomorphic Calculus - arXiv
Nov 8, 2024 · We review the multivariate holomorphic functional calculus for tuples in a commutative Banach algebra and establish a simple naïve extension.
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[2412.04823] Noncommutative complex analytic geometry of ... - arXiv
Dec 6, 2024 · ... commutative modulo its ... noncommutative Frechet algebra presheaf, Harte spectrum, Taylor spectrum, noncommutative holomorphic functional calculus.
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Non-commutative Function Theory and Free Probability - EMS Press
May 3, 2024 · Arens–Calderón–Waelbroeck) holomorphic functional calculus in the Banach al- gebra B ... Let X1,...,Xn be non-commutative random variables in a non-commutative ...